All the relevant observables can be extracted from this formula. Prompt-ring down modes correspond to absence of the (ingoing) term \(e^{-2i\sqrt{\beta z}}\)  which is equivalent to the quantisation of the cycle \(a_D-a\) . In practice one has to compute the quantum period \(a(u)\) , invert it and use quantum Matone relation [1]}, [2]} to find
\({\cal F}(a,q, m_f; \hbar )\)  and then \(a_D= {-1\over 2\pi i} \partial {\cal F}/\partial a\) . Expressing \(a_D\)  and \(a\)  in terms of \(u\) , that in turn is given in terms of \(u_A\)  and \(\tilde{\omega }L\) , one finds an eigenvalue equation for \(\omega _{QNM}\) . The numerical results are collected in appendix .
